Question: Given $ \overrightarrow{BA}\perp\overrightarrow{BD}$, $ m \angle CBD = 2x + 31$, and $ m \angle ABC = 8x + 49$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Explanation: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Since we are given that $\overrightarrow{BA}\perp\overrightarrow{BD}$ , we know ${m\angle ABD = 90}$ Substitute in the expressions that were given for each measure: $ {8x + 49} + {2x + 31} = {90}$ Combine like terms: $ 10x + 80 = 90$ Subtract $80$ from both sides: $ 10x = 10$ Divide both sides by $10$ to find $x$ $ x = 1$ Substitute $1$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 2({1}) + 31$ Simplify: $ {m\angle CBD = 2 + 31}$ So ${m\angle CBD = 33}$.